SigmaCalc

Odds Ratio Calculator

The standard measure of association in case-control studies, comparing odds rather than raw risk.

Exposed, with event (a)

Exposed, without event (b)

Unexposed, with event (c)

Unexposed, without event (d)

Odds ratio

3.857

OR = (a × d) ÷ (b × c)

What you entered

Exposed, with event (a): 30Exposed, without event (b): 70Unexposed, with event (c): 10Unexposed, without event (d): 90

1
Odds of the event in the exposed group
30 ÷ 70= 0.4286
2
Odds of the event in the unexposed group
10 ÷ 90= 0.1111
3
Odds ratio
(30 × 90) ÷ (70 × 10)= 3.8571

Result

Odds ratio: 3.857

An odds ratio of 3.857 means the odds of the event are 3.86× higher in the exposed group compared to the unexposed group.

Odds ratio from a two-by-two table

With a exposed cases, b exposed non-cases, c unexposed cases and d unexposed non-cases, the odds ratio is ad divided by bc. Relative risk is shown alongside for comparison.

a	b	c	d	Odds ratio	Relative risk
30	70	20	80	1.714	1.500
40	60	20	80	2.667	2.000
15	85	10	90	1.588	1.500
50	50	25	75	3.000	2.000

The odds ratio always sits further from 1 than the relative risk, and the gap widens as the outcome becomes more common - compare row three, where a rare outcome gives 1.588 against 1.500, with row four, where a common one gives 3.000 against 2.000. This is why odds ratios from studies of frequent outcomes are routinely misread as relative risks and overstate the effect. An odds ratio of 1 means no association; above 1 means the exposure is associated with higher odds. Association is not causation, and neither figure carries a confidence interval on its own.

Odds ratio vs. relative risk

Odds ratio compares the ratio of odds (event ÷ non-event) between groups, while relative risk compares raw probabilities directly. They converge for rare events but diverge substantially when the event is common — odds ratio is standard for case-control studies where relative risk can't be directly estimated.

Reading the number

An odds ratio of 1 means no association between exposure and the event. Above 1 means the exposed group has higher odds of the event; below 1 means lower odds — how far from 1 indicates the strength of the association.

Frequently asked questions

In a study, 30 of 100 smokers got lung disease vs 5 of 100 non-smokers. What's the odds ratio?

Smoker odds: 30/70 = 0.429. Non-smoker odds: 5/95 = 0.053. OR = 0.429/0.053 = 8.14. Smokers have 8× the odds of developing lung disease compared to non-smokers.

How is odds ratio different from relative risk?

Relative risk compares probabilities directly: 30% vs 5% = RR of 6.0. Odds ratio compares odds: 30/70 vs 5/95 = OR of 8.14. They diverge when the event is common (>10%). For rare events, OR ≈ RR. OR is used in case-control studies; RR in cohort studies. See the relative risk calculator.

What does an odds ratio of 1 mean?

No association — the exposure doesn't change the odds of the event. OR > 1 means increased odds with exposure (risk factor). OR < 1 means decreased odds (protective factor). The 95% confidence interval matters: if it includes 1, the association isn't statistically significant.